Described in provisional U.S. patent application No. 61/290,645 and related U.S. patent application Ser. No. 12/979,778 are dynamically gatable structures called dynamical topology changing (DTC) devices which support an Ising state within arbitrary regions of a fixed curved surface Σ. A material “X” which supports, where appropriately gated, a pure Ising TQFT is detailed in the related applications. Possible local geometries for Σ are shown in FIGS. 1A and 1B hereof. FIGS. 1A and 1B each support a qubit.
It is known from the mathematical work of Bravyi and Kitaev as described in Freedman, Nayak, and Walker [arXiv:cond-mat/0512066], incorporated herein by reference, and Freedman, Nayak, and Walker [arXiv:cond-mat/0512072], incorporated herein by reference, that in the Ising state with (1) sufficiently general manipulations of topology, called smooth cobordism, together with (2) the ability to interferometrically measure topological charge along any simple closed curve on a surface in State X, a topologically protected universal gate set for quantum computing is achieved using Majorana fermions. The resulting computational logical gate set contains (and is generated by) the π/8-phase gate (T), the Hadamard gate (H), and the controlled-Z gate (CZ), represented by:
      T    =          [                                    1                                0                                                0                                              ⅇ                              i                ⁢                                                                  ⁢                                  π                  /                  4                                                                        ]        ,          ⁢      P    =                  T        2            =              [                                            1                                      0                                                          0                                      i                                      ]              ,          ⁢      H    =                  1                  2                    ⁡              [                                            1                                      1                                                          1                                                      -                1                                                    ]              ,          ⁢      CZ    =          [                                    1                                0                                0                                0                                                0                                1                                0                                0                                                0                                0                                1                                0                                                0                                0                                0                                              -              1                                          ]      